Identical representation functions of linear forms

Csaba S\'andor; S\'andor Kiss

REVIEW 2 major objections 3 minor 1 cited by

Identical representation functions of linear forms

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that the weighted representation function of linear forms determines both the underlying multiset and the coefficient vector, and classifies nontrivial coincidences.

desk verdict Genuine extension of Nathanson's theorem; a repairable counting gap in Lemma 2.1(ii) blocks Theorem 1.6 as written. read the letter →

arxiv 2506.03983 v1 pith:IK76INMS submitted 2025-06-04 math.NT

Sándor Kiss , Csaba Sándor This is my paper

classification math.NT MSC 11B34

keywords additivenumbertheoryrepresentationfunctionlinearformsmultisetsuniquenessofprimitivesolutionsdigitsetslogarithmicindependence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends the classical theorem that the binary representation function determines a set of nonnegative integers to representation functions attached to linear forms. Its subject is the count $R_{A,\alpha}(n)$ of ways to write $n$ as $\alpha_1 a_{j_1}+\cdots+\alpha_s a_{j_s}$ with $a_j$ drawn from a multiset $A$. The central claim is rigidity: for multisets with at least two elements, equality of these counts forces $A=B$ when the coefficient vectors agree, and forces $\alpha=\beta$ when the multisets agree. The paper also classifies how shifts of $A$ and $B$ can preserve equality, reduces every nontrivial equality to a primitive one, and constructs primitive examples whose behavior is controlled by whether the logarithms of two bases are rationally independent. If the claim is correct, the weighted representation function acts as a unique fingerprint for both the set and the weights.

What carries the argument

The central object is the weighted multiset $\alpha A=\alpha_1A+\cdots+\alpha_sA$, because $R_{A,\alpha}(n)$ is exactly the characteristic function $\chi_{\alpha A}(n)$: two representation functions agree precisely when the multisets $\alpha A$ and $\beta B$ coincide. The proof of rigidity is carried by Lemma 2.1, a three-part induction principle comparing initial segments of multisets and coefficient vectors: if the weighted multisets and the coefficient vectors agree up to a carefully chosen threshold, then the underlying objects agree one step further, so a hypothetical smallest disagreement is impossible. The structural results also use the Kronecker product $\alpha\beta$, which encodes the factorization identity $(\alpha\beta)A=\alpha(\beta A)$ and supplies the simplification process in Theorem 1.11.

What would settle it

Take $\beta=\alpha=(1,2)$, $D=C=\{0,1\}$, and $k=2$ in Lemma 2.1(ii). The lemma's hypotheses hold trivially, but the decomposition used in the proof gives left side $R_{C,\alpha}(2)=1$ (from $2=2\cdot1$) and right side $0$, because the representation using the second coefficient is not counted; this shows the proof of the lemma, and hence of Theorem 1.6(i), misses a representation class that must be handled.

Watch

Extended reading notes

Core claim

The paper's central assertion is that the equation $R_{A,\alpha}=R_{B,\beta}$ is rigid. For $A,B\in\mathcal{N}_m$ with $|A|,|B|\ge 2$, it proves that $\alpha A=\alpha B$ implies $A=B$ and $\alpha A=\beta A$ implies $\alpha=\beta$; in particular a nontrivial coincidence requires changing both the multiset and the coefficient vector. It further shows that if shifted multisets $A+u$, $B+v$ have equal weighted representation functions, then the shifts are forced to be $u=w(\sum\beta_i)/g$ and $v=w(\sum\alpha_i)/g$ for a common integer $w$, where $g=\gcd(\sum\alpha_i,\sum\beta_i)$. Any solution can be simplified to a primitive solution $\gamma C=\delta D$ that cannot be produced by the elementary factorization construction, and for the explicit second construction built from digit sets $S(\lambda,N)$, being primitive is equivalent to not following from the first construction and equivalent to $\log\lambda,\log\lambda'$ being linearly independent over $\mathbb{Q}$. The paper thus reduces the open classification problem to finding all primitive solutions beyond the constructions it exhibits.

Load-bearing premise

The induction in Lemma 2.1(ii) assumes that every representation of $\alpha_1 k$ using a coefficient at least $k$ is of the special form using one copy of the smallest coefficient $\alpha_1$ together with the element $k$, so that no representation involving a different large coefficient together with a nonzero element of $C$ is missed.

Editorial extensions

If this is right

With a fixed coefficient vector, the weighted representation function determines the multiset uniquely: $R_{A,\alpha}=R_{B,\alpha}$ implies $A=B$ whenever $|A|,|B|\ge 2$.
With a fixed multiset, the representation function determines the coefficient vector uniquely: $R_{A,\alpha}=R_{A,\beta}$ implies $\alpha=\beta$.
A nontrivial coincidence $R_{A,\alpha}=R_{B,\beta}$ must change both arguments, and shifted coincidences are governed by the single integer $w$ in Theorem 1.4.
Every coincidence can be decomposed to a primitive coincidence $\gamma C=\delta D$, so the classification of primitive solutions is the complete remaining task.
In the digit-set family, the primitive, non-first-construction solutions are exactly those with $\log\lambda$ and $\log\lambda'$ linearly independent over $\mathbb{Q}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One testable strengthening suggested by the proof is that equality on a sufficiently long initial segment already forces equality of truncated multisets and coefficient vectors, since the induction only compares finite windows; this would make the rigidity decidable from finite data.
The logarithmic criterion suggests that further primitive examples, if they exist, should be sought among multiplicatively independent bases, since commensurable logarithms force the second construction to collapse into the first.
If the missed representation class in Lemma 2.1(ii) is not excluded by the lemma's hypotheses, the rigidity theorem may still be true with a refined decomposition that counts large-coefficient pairs explicitly; checking that refined count is the natural repair.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: this paper genuinely extends Nathanson's theorem to linear forms and multisets, and the dichotomy in Theorem 1.13 relating primitive solutions to Q-linear independence of logarithms is a real insight. But the proof of the central uniqueness theorem has a hole. Lemma 2.1(ii) contains a false decomposition, and until that is corrected Theorem 1.6(i) is not established as written.

What's good: the problem is natural and the paper does not oversell. Theorem 1.6, if it stands, is the right generalization of Nathanson's 1978 result. The first construction (alpha1 alpha2 = beta1 beta2) is correct, and the authors are explicit that iterating it doesn't give anything new. The reduction to primitive solutions in Theorem 1.11 is intricate and appears to limit the vector splittings in a bounded way. The second construction, using base-lambda representations with N factored into blocks, is clever, and Theorem 1.13 spells out when those examples are genuinely primitive. The paper also responsibly lists open problems.

The soft spot is the one the stress-test identifies. In Lemma 2.1(ii), the proof asserts

R_{C,alpha}(alpha1 k) = |{(j_1,...,j_K) : alpha1 k = sum_i alpha_i c_{j_i}, c_{j_i}

That identity is false. With alpha=(1,2), C={0,1}, k=2, the representation using alpha_2=2 and the element 1 gives alpha1 k = 2, and it is counted by neither term. The RHS is 0, the LHS is 1. The missing class involves chi_alpha(k) chi_C(1). Since the proof then cancels the displayed second terms, the omitted class could change the conclusion. I don't see that the lemma's hypotheses prevent such terms; they may cancel across C and D because of minimality of k, but the paper doesn't show it. This is a genuine gap, not a stylistic issue. The statement of Lemma 2.1(ii) might well be true, but the submitted proof doesn't establish it.

One smaller caveat: Theorem 1.13 leans on the authors' prior classification in [7] as a black box. That's acceptable if [7]'s exact theorem is stated; the paper just cites it, which makes verification harder. The proof of Theorem 1.11 is dense and I did not check every line; it looks plausible but should receive careful referee attention.

Who this is for: additive number theorists working on representation functions. The construction of primitive solutions is the most valuable part and would survive even if Theorem 1.6 needed a different proof. As submitted, it shouldn't be accepted: the false count in Lemma 2.1(ii) is load-bearing. But it deserves a serious referee and a major revision, not a desk reject. I'd be willing to see it again once the missing terms are accounted for.

Referee Report

2 major / 3 minor

Summary. The paper studies the representation function R_{A,α}(n) that counts tuples (j_1,...,j_s) with Σ α_i a_{j_i} = n for a multiset A of nonnegative integers and a positive integer vector α. The main claims are: Theorem 1.6, that for |A|,|B|≥2, αA = αB implies A=B and αA = βA implies α=β; Theorem 1.11, that every equality αA = βB can be simplified to a primitive solution; and Theorem 1.13, which characterizes the constructed second-type solutions by linear independence of logarithms. The central tool is Lemma 2.1, a local uniqueness statement that compares truncated coefficient vectors and truncated multisets from truncated equality of αC and βD.

Significance. If the results are correct, they would give a natural and substantial extension of Nathanson's theorem to weighted linear forms and to multisets, with a clean uniqueness statement (Theorem 1.6), a reduction-to-primitive framework (Theorem 1.11), and explicit families of non-trivial primitive solutions with a logarithm-independence criterion (Theorem 1.13). The paper is largely self-contained and makes honest use of the authors' prior published classification result [7] in the proof of Theorem 1.13. However, the central lemma is not proven as written, so the significance is conditional on a repair of that proof.

major comments (2)

[Section 2, Lemma 2.1(ii)] The displayed decomposition of R_{C,α}(α1 k) in the proof of Lemma 2.1(ii) is false. For α=(1,2), C={0,1}, and k=2, we have α1 k = 2 and K=1. The right-hand side is |{(j1): 2 = 1·c_{j1}, c_{j1}<2}| + χ_α(1)χ_C(2) = 0 + 1·0 = 0. But the left-hand side is 1, because (j1,j2)=(1,1) gives 1·0 + 2·1 = 2. The missing representation uses a coefficient α_i ≥ k with an element c_j < k; in this example α_2 = k = 2 and c_j = 1. The proof's claim that this case is 'similar to the previous case' is not valid: in part (i) the target kc2 forces any positive element paired with a coefficient ≥ k to be exactly c2, whereas in part (ii) the target α1 k admits many combinations of coefficients ≥ k with elements in {1,...,k−1}. This is not a minor typo: Lemma 2.1(ii) is the key ingredient in the proof of Theorem 1.6(i) and is also used in the proof of Theorem 1.11. The lemma statement may be true and the proof may be repairable, but the submitted argument does not establish it.
[Sections 4 and 5 (Theorems 1.6 and 1.11)] Both Theorem 1.6(i) and Theorem 1.11 depend directly on Lemma 2.1(ii): Theorem 1.6(i) applies the lemma with α=β, and Theorem 1.11 uses it repeatedly to pass from truncated equality of factorized multisets to equality of C^(l) ∩ [n]_m. Because the lemma's proof is invalid, the main uniqueness statement and the reduction theorem are not established as written. The paper should either supply a correct proof of Lemma 2.1(ii), including a full accounting of representations with coefficients ≥ k and elements < k, or restructure the argument to avoid this step.

minor comments (3)

[Corollary 1.7] Corollary 1.7 refers to 'Proposition 1.4' but the relevant statement is Theorem 1.4.
[Section 1, definition of primitive solution] The definition of a primitive solution is difficult to parse; the phrase 'whenever α1 = β1 = α2 = β2 = (1)' should presumably be 'then α1 = β1 = α2 = β2 = (1)', and the intended meaning (no nontrivial factorization) should be stated explicitly.
[Proof of Theorem 1.11, first step] The text 'there exist a natural number in and for with' should read 'there exists a natural number i_n such that'; the indexing of i_n is otherwise unclear.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.6 is proved from a counting lemma, not from fitted inputs; the sole self-citation ([7]) is an independent published classification used as a lemma.

full rationale

The main derivation is self-contained in the relevant sense. The equality R_{A,alpha}(n) = chi_{alpha A}(n) is definitional (Eq. (1)), and the paper then proves structural uniqueness statements (Theorems 1.6 and 1.11) by a counting lemma (Lemma 2.1) and algebraic factorization arguments. No parameter is fitted to data, and no 'prediction' is a renamed input: the equal-representation-function condition is equivalent by definition to equality of multisets alpha A = beta B, and the theorems classify when that can happen. The only self-citation in a load-bearing position is Theorem 1.2 of [7], used in the proof of Theorem 1.13(i)->(ii). That cited theorem is a published classification of unique representations by linear forms; it supplies a structural fact about coefficient vectors when a representation function is identically 1, with assumptions distinct from the conclusion being proved here. It is not a parameter fitted to the present data and it does not assert the equivalence Theorem 1.13 is proving, so under the stated rules it counts as independent support and does not raise the circularity score. A reviewer's counterexample to an identity inside the proof of Lemma 2.1(ii) indicates a possible gap in the written proof, but a proof gap is a correctness concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

No fitted parameters or invented entities: lambda, N, u, v are free construction variables, not parameters fitted to data. The only significant external input is the classification theorem from the authors' earlier paper [7], which is cited as a black box.

assumptions (2)

domain assumption N_m is the class of multisets of nonnegative integers with exactly one 0 and finite multiplicities; all A, B in the theorems are assumed to lie in N_m.
Defined in Section 1 before Problem 1.3; the main theorems apply only to this class.
domain assumption Theorem 1.2 of [7] (Kiss and Sandor, Unique representations of integers by linear forms): if R_{C,alpha}(n)=1 for all n, then the coefficient vector is a geometric progression in a base lambda_0.
Used as a black box in the proof of Theorem 1.13 (Section 6) to infer the structure of gamma, mu, nu when a representation function is identically 1.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Identical representation functions of linear forms." pith.science (2026). https://pith.science/paper/IK76INMS

@misc{pith2026250603983,
  author       = {Pith},
  title        = {Pith review of: Identical representation functions of linear forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IK76INMS}},
  note         = {Machine review of arXiv:2506.03983}
}

abstract

For a set of natural numbers $A$, let $R_{A}(n)$ be the number of representations of a natural number $n$ as the sum of two terms from $A$. Many years ago, Nathanson studied the conditions for the set $A$ and $B$ of natural numbers that are needed to guarantee that $R_{A}(n) = R_{B}(n)$ for every positive integer $n$. In the last decades, similar questions have been studied by many authors. In this paper, we extend Nathanson's result to representation functions associated to linear forms and we study related problems.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Flexible Catalysis
quant-ph 2025-10 conditional novelty 8.0 of 10

Flexible catalysis—allowing a catalyst to transform into another valid catalyst—strictly increases which bipartite quantum state extractions are possible under local unitaries and permutation matrices, and generalizes...

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · cited by 1 Pith paper

[7]

S. Z. Kiss, Cs. S´andor. Unique representations of integers by linear forms, Journal of Combin. Theory Ser. A, 213 (2025), 106007

work page 2025
[1]

Y-G. Chen, M. Tang. Partitions of natural numbers with the same representation functions, J. Number Theory 129 (2009), 2689-2695

work page 2009
[2]

Chen, V.F

Y-G. Chen, V.F. Lev. Integer sets with identical representation functions, Integers 16 (2016), A36

work page 2016
[3]

Y-G. Chen, B. W ang. On additive properties of two special sequences, Acta Arith. 110 (2003), 299-303

work page 2003
[4]

G. Dombi. Additive properties of certain sets, Acta Arith. 103 (2002), 137-146

work page 2002
[5]

S. Z. Kiss, Cs. S ´andor. On the structure of sets which have coinciding represen- tation functions , Integers 19 (2019), A66, 29 pp

work page 2019
[6]

S. Z. Kiss, Cs. S ´andor. Partitions of the set of nonnegative integers with the same representation functions , Discrete Math. 19 (2019), A66, 29 pp

work page 2019
[8]

S.Z. Kiss, E. Rozgonyi, Cs. S´andor. Sets with almost coinciding representation functions, Bull. Aust. Math. Soc. 89 (2014), 97-111

work page 2014

Show all 14 references

[9]

M. B. Nathanson. Representation functions of sequences in additive number the- ory, Proc. Amer. Math. Soc. 72 (1978), 16-20. 19

1978
[10]

Rozgonyi, Cs

E. Rozgonyi, Cs. S ´andor. An extension of Nathanson ’s Theorem on representa- tion functions , Combinatorica, 164 (2016), 1-17

2016
[11]

S ´andor

Cs. S ´andor. Partitions of natural numbers and their representation functions , In- tegers 4 (2004), A18, 5 pp

2004
[12]

M. Tang. Partitons of the set of natural numbers and their representation functions, Discrete Math. 308 (2008), 2614-2616

2008
[13]

M. Tang, W. Yu. A note on partitions of natural numbers and their representation functions, Integers 12 (2012), A53

2012
[14]

X.-H. Yan. Partitions of the set of nonnegative integers with identical representa- tion functions , Int. J. Number Theory 15 (2019), 1969-1975. 20

2019

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [7]

S. Z. Kiss, Cs. S´andor. Unique representations of integers by linear forms, Journal of Combin. Theory Ser. A, 213 (2025), 106007

work page 2025

[2] [1]

Y-G. Chen, M. Tang. Partitions of natural numbers with the same representation functions, J. Number Theory 129 (2009), 2689-2695

work page 2009

[3] [2]

Chen, V.F

Y-G. Chen, V.F. Lev. Integer sets with identical representation functions, Integers 16 (2016), A36

work page 2016

[4] [3]

Y-G. Chen, B. W ang. On additive properties of two special sequences, Acta Arith. 110 (2003), 299-303

work page 2003

[5] [4]

G. Dombi. Additive properties of certain sets, Acta Arith. 103 (2002), 137-146

work page 2002

[6] [5]

S. Z. Kiss, Cs. S ´andor. On the structure of sets which have coinciding represen- tation functions , Integers 19 (2019), A66, 29 pp

work page 2019

[7] [6]

S. Z. Kiss, Cs. S ´andor. Partitions of the set of nonnegative integers with the same representation functions , Discrete Math. 19 (2019), A66, 29 pp

work page 2019

[8] [8]

S.Z. Kiss, E. Rozgonyi, Cs. S´andor. Sets with almost coinciding representation functions, Bull. Aust. Math. Soc. 89 (2014), 97-111

work page 2014